\newproblem{lay:4_3_25}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 4.3.25}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Jan. 20th, 2013} \\}{}

  % Problem statement
	Let $\mathbf{v}_1=(1,0,1)$, $\mathbf{v}_2=(0,1,1)$ and $\mathbf{v}_3=(0,1,0)$, and let $H$ be the set of vectors of $\mathbb{R}^3$ whose second and 
	third entries are equal. Then every vector in $H$ has a unique expansion as a linear combination of $\mathbf{v}_1$, $\mathbf{v}_2$ and $\mathbf{v}_3$
	because $(s,t,t)=s(1,0,1)+(t-s)(0,1,1)+s(0,1,0)$. Is $\left\{\mathbf{v}_1,\mathbf{v}_2,\mathbf{v}_3\right\}$ a basis of $H$? Why or why not?
}{
  % Solution
	$\mathbf{v}_1$ and $\mathbf{v}_3$ do not belong to $H$ (because they
	do not have the same values in the second and third position), so they cannot participate of any basis of $H$.
}
\useproblem{lay:4_3_25}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
